Tangent Theorem 1

In this post, we discuss a theorem on tangent of circles.  We show that if a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. In the figure below, line m is perpendicular to $OA$ at $A$. We show that it is a tangent to circle O.

Theorem

If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle.

Given

Line $m$ is perpendicular to $OA$ at $A$.

Prove

Line $m$ is tangent to circle $O$

Proof

Let $B$ be another point on line $m$ other than $A$.

Since $OA$ is perpendicular $m$, triangle $OAB$ is a right triangle with hypotenuse $OB$. This means that $OB > AB$ and $B$ must be on the exterior of the circle. Therefore $B$ cannot lie on the circle and $B$ is the only point of m that is on the circle. So, $m$ is a tangent to the circle.

The converse of this statement is also true (see proof).